1. Field
This invention relates to instructional visual aid devices which provide visual transformation of superimposed picture images over prescribed stationary pictures to aid in proving most of the theorems in a geometry course, using the method of deductive reasoning. These manipulatives can also be used to display visually geometric problem solving, mathematical concepts, or algebraic linear graphs for which a mathematical system can be found.
2. State of the Art
Overlay transparencies are widely used in education on overhead projectors for projecting a superimposed picture image over a stationary transparency. They are also used in direct demonstration without an overhead projector by overlaying a transparency on a flat opaque background surface. Many types of superimposed visual aid devices are known, such as those disclosed by U.S. Pat. Nos. 3,556,397; 3,827,163; 4,655,714; and 4,705,478. The disclosed U.S. Pat. No. 4,705,478 uses superimposed feature devices to demonstrate visually two theorems in geometry, where a geometrical construction is applied to locate a unique location for the entre of rotation in order to guarantee the superimposed feature to assume a desired location. The mechanism of operating the referred disclosed U.S. Pat. No. 4,705,478 is cumbersome, in that two members move in a curved path over the surface of a diagram and curved slots confine the pivotal movements to stop at the desired position. In the article of the disclosed patent, U.S. Pat. No. 4,705,478, the finished product does not show the circle circumscribed about triangle ABC, or the tangent CD of the circle at point C, the point of tangency. The circle and the tangent are essential features so that the device may agree with the methods of deductive reasoning in proving the theorem. However, the device appears to demonstrate that the sum of the angles of a triangle equals 180.degree. using superimposed means in transforming two angles congruent to angles A and B to form with angle C a straight angle. The superimposed feature must have the diagram of the circumscribed circle and the tangent to the circle as a background so that the theorem can be supported by geometric logic. Consequently, the techniques applied in the disclosed patent device suggest the use of informal geometry. In the new invention, a prescribed entre of rotation that guarantees a superimposed feature is more feasible, advantageous, preferrable and practical, since many geometric theorems can be designed for visual aids, rather than using geometric construction to locate one or more entres of rotation for a prescribed diagram as applied in the disclosed U.S. Pat. No. 4,705,478. The features of U.S. Pat. Nos. 3,556,397; 3,827,163; and 4,655,714 are to display visually the values of the trigonometric functions, but not the behavior of their graphs.
The fundamental concept of the claimed present invention is to apply the transformation of a set of points, figures, or linear graphs such that the set of points of a figure or a line are symmetric with respect to the entre of rotation. Additionally, the devices demonstrate transformation of a picture image with respect to a picture through an axis of reflection and transformation of an image picture guided by two prescribed parallel lines. In this invention, the manipulatives do not always necessitate a set of points of a figure to be symmetric relative to the entre of rotation. Neither are the points always symmetric with respect to the line of reflection. The forthcoming article of the summary of this invention will provide concrete tangible applications for the claimed invention. The components of the devices of this invention are simple, easy to operate, long lasting, and inexpensive to produce. Thus, the basic advantage of this invention is its versitility of application(s) to almost all theorem proofs in a standard secondary geometry course. Previous patented inventions address a very limited scope of theorem applications. The same learning devices described have additional applications in illustrating algebraic graphs.